ARTICLE pubs.acs.org/Langmuir
On the Behavior of Dew Drops Martin E. R. Shanahan* Universite de Bordeaux, Institut de Mecanique et Ingenierie de Bordeaux (I2M) - UMR, CNRS 5295, B^atiment A4, 351 Cours de la Liberation, 33405 TALENCE Cedex, France ABSTRACT: It may be observed that, when dew drops form, although they may be positioned randomly on flat leaves, they tend to accumulate at the pointed ends of thin, slightly conical growths. We discuss here the basic physics leading to this phenomenon.
’ INTRODUCTION At the recent 85th American Chemical Society (ACS) Colloid and Surface Science Symposium (Montreal, Quebec, Canada), in a convivial conversation with T. Thundat (University of Alberta, Canada), E. Bonaccurso (Technical University Darmstadt, Germany), and H.-J. Butt (Max Planck Institute, Mainz, Germany), it was evoked that dew drops on fine or spindly leaves often tend to accumulate at the leaf tip, whereas on flat leaves, they seem to show no preference for their (apparent) equilibrium position. Although no definite conclusion was reached during the discussion, it is this exchange that was the instigation of the following contribution. Clearly, the random positioning of dew drops on flat leaves may be expected; any slight wetting hysteresis should be sufficient to prevent the descent of small drops along the substrate because of gravity, even if vertical. In the majority of cases, such drops are of dimensions comparable to or smaller than the capillary length, k1 = (γ/Fg)1/2, where γ, F, and g are liquid surface tension, density, and gravitational acceleration (ca. 2.8 mm for water), respectively. However, the behavior of droplets terminating at the apex of what amounts to conical structures of very small angle is more intriguing. The actual mechanism of depositing of liquid from the vapor is not considered here. To keep the treatment fairly simple, we assume that the solid is an axisymmetric cone of small apex half-angle α (,1) and that the liquid will appear after condensation, as either a thin film (Figure 1) (but not thin enough for long-range forces to become significant) or a small droplet (Figure 2), respecting symmetry with respect to the cone axis, and at some distance from the apex. In addition, the contact angle of the liquid, θ, on the substrate will be taken to be less than π/2. This simplification should limit effects, such as “rolling up” into “clam shapes”, known for droplets of large contact angle on (cylindrical) fibers.1 Liquid tends to accumulate toward one side of the cylinder because of instability of the axisymmetric configuration.16 Although we treat the solid as a cone, a necessary condition for the effect in question, the low apex angle could easily lead to such “clam” conformations at high θ. The r 2011 American Chemical Society
tendency for this asymmetry is admittedly marked for θ > π/3 but is beyond the scope of this na€ive treatment.
’ THIN FILM Adopting Cartesian coordinates, (x, z), with x following the axis of symmetry and z being the radial distance of the liquid/ vapor interface, and the cone apex being at the origin, we assume the presence of a thin, homogeneous film of liquid on the solid because of condensation, of thickness ε cos α ≈ ε and covering the cone over x1 e x e x2, representing a volume of revolution (Figure 1; angle α is exaggerated for clarity). With the equations representing the cone surface, r, and that of the upper liquid surface, z, being simply r = x tan α and z = x tan α + ε, the volume, V, of the liquid (constant, assumed not to exchange with the environment) is simply V ¼π ≈
Z x 2 x1
½ðx tan α þ εÞ2 x2 tan2 αdx
πεðr2 r1 Þ πε ½ðr2 þ r1 Þ þ ε ≈ ðr2 2 r1 2 Þ tan α tan α ð1Þ
Small end effects near the triple line are neglected. The “skin” area, A, corresponding to the exposed liquid/environment interface is given by A ¼ 2π
Z x2 x1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi π 1 þ tan2 αðr2 r1 Þ z 1 þ z0 2 dx ≈ ½ðr2 þ r1 Þ þ 2ε tan α
π ≈ ðr2 2 r1 2 Þ sin α
ð2Þ
where z0 = dz/dx. Because it is assumed that (r2 + r1) . ε, eq 2 is equally valid for the interfacial area between the substrate and liquid (in fact, the Received: August 23, 2011 Revised: October 21, 2011 Published: November 07, 2011 14919
dx.doi.org/10.1021/la203316k | Langmuir 2011, 27, 14919–14922
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thicken in the z direction. A second scenario is that there may indeed be a tendency for the film to thicken to become a drop rather than a sheath. We now consider this aspect.
Figure 1. Schematic representation of a thin film of liquid on a circular, conical substrate.
’ AXISYMMETRIC DROP In Figure 2, we represent schematically an axisymmetric drop on a cone of low semi-angle, α. This type of problem has been studied previously for fibers13,5,912 and also more recently on a conical wire.13 Again, we employ Cartesian coordinates to describe the drop, having assumed contact angles less than ca. π/2 [in fact,
